Level of repair analysis and minimum cost homomorphisms of graphs

Level of Repair Analysis and Minimum CostHomomorphisms of Graphs

Gregory Gutin1, Arash Rafiey1, Anders Yeo1, and Michael Tso2

1 Department of Computer ScienceRoyal Holloway University of London

Egham, Surrey TW20 OEX, UKgutin,arash,[email protected]

2 School of Mathematics, University of ManchesterP.O. Box 88, Manchester M60 1QD, UK

[email protected]

This paper is dedicated to the memory of Lillian Barros

Abstract. Level of Repair Analysis (LORA) is a prescribed procedure for defence logisticssupport planning. For a complex engineering system containing perhaps thousands of assem-blies, sub-assemblies, components, etc. organized into several levels of indenture and with anumber of possible repair decisions, LORA seeks to determine an optimal provision of repairand maintenance facilities to minimize overall life-cycle costs. For a LORA problem with twolevels of indenture with three possible repair decisions, which is of interest in UK and USmilitary and which we call LORA-BR, Barros (1998) and Barros and Riley (2001) developedcertain branch-and-bound heuristics. The surprising result of this paper is that LORA-BRis, in fact, polynomial-time solvable. To obtain this result, we formulate the general LORAproblem as an optimization homomorphism problem on bipartite graphs, and reduce a gen-eralization of LORA-BR, LORA-M, to the maximum weight independent set problem on abipartite graph. We prove that the general LORA problem is NP-hard by using an importantresult on list homomorphisms of graphs. We introduce the minimum cost graph homomor-phism problem, provide partial results and pose an open problem. Finally, we show thatour result for LORA-BR can be applied to prove that an extension of the maximum weightindependent set problem on bipartite graphs is polynomial time solvable.

Keywords: Computational Logistics; Level of Repair Analysis; Independent Sets in Graphs;Homomorphisms of Graphs

1 Introduction

Level of Repair Analysis (LORA) is a prescribed procedure for defence logistics supportplanning (see, e.g., Crabtree and Sandel [10] and the website of the UK MoD AcquisitionManagement System at www.ams.mod.uk/ams). For a complex engineering system con-taining perhaps thousands of assemblies, sub-assemblies, components etc. organized into` ≥ 2 levels of indenture and with r ≥ 2 possible repair decisions, LORA seeks to deter-mine an optimal provision of repair and maintenance facilities to minimize overall life-cyclecosts.

Barros [4] and Barros and Riley [6] provide a generic integer programming formulationof the LORA optimization problem for systems with ` levels of indenture and r possible re-pair decisions (including the non-repair option). A special case with ` = 2 and r = 3, whichwe call LORA-BR, is of particular importance because it corresponds to recommendationsin certain UK and US military standard handbooks, see Barros and Riley [6]. In Frenchmilitary standards, ` = 2 and r = 5. Notice that the actual research of Barros and Rileywas only for LORA-BR [5] for which the corresponding software have been developed.

While Barros [4] solves LORA-BR using a general purpose IP solver, Barros and Riley[6] outline a specialized branch-and-bound heuristic, which appears to be more efficientin computational experiments. Their heuristic is based on a relaxation of LORA-BR intoa pair of uncapacitated facility location (UFLP) problems [9, 14]. A branch-and-boundprocedure then employs local search heuristics to satisfy additional side constraints ensuringconsistency between repair decisions for pairs of items nested on adjacent indenture levels.Since UFLP is NP–hard [9, 13, 14], it could be expected that LORA-BR would also beintractable. However, the surprising result of this paper is that LORA-BR is polynomiallysolvable and this is achieved by reducing its generalization, LORA-M (defined in Section3), to the maximum weight independent set problem on a bipartite graph.

As it was pointed out above, the case of two levels of indenture is of particular interest(e.g., in UK, USA and French military). For clarity of exposition, in the rest of this paperapart from Section 4, we restrict ourselves to two levels of indenture, ` = 2, but ourapproach can be extended to arbitrary ` as demonstrated in Section 4.

We will use the notion of a homomorphism of graphs that generalizes the notion ofcoloring (see, e.g., Hell and Nesetril [16]). For a pair of graphs H = (V (H), E(H)) andB = (V (B), E(B)), a mapping k : V (B)→V (H) such that if xy ∈ E(B) then k(x)k(y) ∈E(H) is called a homomorphism of B to H. To study the LORA problem, we show howto formulate it as a problem of finding a homomorphism of minimum cost belonging to acertain class of homomorphisms of a bipartite graph to a fixed bipartite graph. This allowsus to use a nontrivial result on the list H-homomorphism problem from [11] to easily showthat the general LORA problem with ` = 2 is NP–hard. We also prove that LORA-M ispolynomial time solvable.

The formulation of the LORA problem in terms of special homomorphisms leads usto the introduction of the minimum cost H-homomorphism problem (MCHP): For a fixedgraph H and an input graph G given together with costs cz(u), the cost of mapping avertex u ∈ V (G) to z ∈ V (H), verify whether there is a homomorphism of G to H, and ifone exists, find such a homomorphism k that minimizes

∑u∈V (G) ck(u)(u). MCHP extends

the well-studied list H-homomorphism problem [16]. We use our results for the LORAproblem to obtain the corresponding results for MCHP. In particular, we show that if H isa bipartite graph with the complement being an interval graph, then MCHP is polynomialtime solvable. In contrast, if H is not bipartite with the complement being a circular arcgraph, then MCHP is NP–hard.

We also use our results to show that the bipartite case of the critical independent setproblem (defined in Section 6), which generalizes the maximum weight independent setproblem, is polynomial time solvable.

In this paper, all graphs are finite, undirected, and simple (i.e., without loops or multipleedges). For standard graph-theoretical terminology and notation, see, e.g., Asratian, Denleyand Haggkvist [3] or West [18]. For terminology and results on homomorphisms, see Helland Nesetril [16].

The rest of the paper is organized as follows. In Section 2, we provide formulations ofLORA-BR and the general LORA problem with ` = 2 in terms of graph homomorphisms.We prove that the general LORA problem with ` = 2 is NP–hard. In Section 3, we showhow to solve a generalization of LORA-BR, LORA-M with ` = 2, in polynomial time. InSection 4, we extend the general LORA problem with ` = 2 to the general LORA problemwith arbitrary ` ≥ 2 as well extend the main result of Section 3. In Section 5, we introducethe minimum cost H-homomorphism problem and show that the results of Sections 2 and3 can be easily extended to it. In the end of the section, we pose an open problem. Finally,in Section 6 we apply a result from Section 3 to solve the bipartite case of the criticalindependent set problem in polynomial time.

2 LORA-BR and General LORA with ` = 2

Consider first a special case of LORA with ` = 2 and r = 3 following Barros [4] andBarros and Riley [6] (we will call this special case LORA-BR). We refer to the first level ofindenture in LORA-BR as subsystems s ∈ S and the second level of indenture as modulesm ∈ M. The distribution of modules in subsystems can be given by a bipartite graphG = (V1, V2;E) with partite sets V1 = S and V2 = M . For arbitrary s ∈ V1 and m ∈ V2,sm ∈ E if and only if module m is in subsystem s. We consider G to be an arbitrarybipartite graph and denote its vertex set V (V = V1 ∪ V2).

There are r = 3 available repair decisions for each level of indenture: ”discard”, ”localrepair” and ”central repair”, labelled respectively D, L,C (subsystems) and d, l, c (mod-ules). To be able to use a decision z ∈ {D, L, C, d, l, c}, we have to pay a fixed cost cz.Assume also known additive costs (over a system life-cycle) cz(u) of prescribing repairdecision z for subsystem or module u.

We wish to minimize the total cost of choosing a subset of the six repair decisions andassigning available repair options to the subsystems and modules subject to the followingconstraints:

If a module m occurs in subsystem s (i.e., sm ∈ E) we impose the following logicalrestrictions on the repair decisions for the pair (s, m) motivated through practical consid-erations:

R1 : Ds ⇒ dm, R2 : lm ⇒ Ls,

where Ds, dm denote the decisions to discard subsystem s, module m, respectively, etc.Notice that even though module m may be common to several subsystems we are requiredto prescribe a unique repair decision for that module.

R1 has the interpretation that a decision to discard subsystem s necessarily entails dis-carding all enclosed modules. R2 is a consequence of R1 and a policy of “no backshipment”which rules out the local repair option for any module enclosed in a subsystem which issent for central repair [6].

Let FBR = (Z1, Z2; T ) be a bipartite graph with partite sets Z1 = {D, C,L} (sub-system repair options) and Z2 = {d, c, l} (module repair options) and with edges T ={Dd,Cd,Cc, Ld, Lc, Ll}. Let Z = Z1 ∪ Z2. Observe that any homomorphism k of G toFBR such that k(V1) ⊆ Z1 and k(V2) ⊆ Z2 satisfies the rules R1 and R2. Indeed, let u ∈ V1,v ∈ V2, uv ∈ E. If k(u) = D then k(v) = d, and if k(v) = l then k(u) = L.

Let Li ⊆ Zi, i = 1, 2. We call a homomorphism k of G to FBR an (L1, L2)-homomorphismof G to FBR if k(u) ∈ Li for each u ∈ Vi, i = 1, 2. Now LORA-BR can be formulated asthe following graph-theoretical problem: We are given a bipartite graph G = (V1, V2; E),V = V1 ∪ V2, and we consider homomorphisms k of G to FBR. (If no homomorphisms ofG to FBR, then the problem has no feasible solution.) Mapping of u ∈ V to z ∈ Z (i.e.,k(u) = z) incurs a real cost cz(u). The use of a vertex z ∈ Z in a homomorphism k (i.e.,k−1(z) 6= ∅) incurs a real cost cz. We wish to choose subsets Li ⊆ Zi, i = 1, 2, and find an(L1, L2)-homomorphism k of G to FBR that minimize

∑

u∈V

ck(u)(u) +∑

z∈L1∪L2

cz. (1)

We call the expression in (1) the cost of k.

The graph-theoretical formulation of LORA-BR can be naturally extended as follows:The above problem with FBR replaced by an arbitrary fixed bipartite graph F = (Z1, Z2; T )is called the general LORA problem with ` = 2. Let Z = Z1 ∪ Z2. Notice that the generalLORA problem with ` = 2 extends the generic formulation of the LORA problem with ` = 2given in [6]. The formulation of the general LORA problem (with arbitrary `) provided inSection 4 extends the generic formulation of the LORA problem (with arbitrary `) givenin [6].

To prove that the general LORA problem with ` = 2 is NP–hard, we will use animportant result on the list H-homomorphism problem defined below. Suppose that weare given a pair of graphs H and B and a list Λ(v) ⊆ V (H) for each v ∈ V (B). Ahomomorphism f : V (B)→V (H) such that f(v) ∈ Λ(v) for each v ∈ V (B) is called aΛ-homomorphism. For a fixed H, the list H-homomorphism problem asks whether thereexists a Λ-homomorphism f of B to H for an input graph B with lists Λ.

A graph P = (V (P ), E(P )) is a circular arc graph if there is a family of arcs Av,v ∈ V (P ), on a fixed circle, such that xy ∈ E(P ) if and only if Ax and Ay intersect. Feder,Hell and Huang [11] obtained the following important result.

Theorem 1. If H is a bipartite graph with the complement being a circular arc graph, thenthe list H-homomorphism problem is polynomial time solvable. Otherwise, the problem isNP–complete.

Observe that, if H is bipartite, we may restrict inputs B of the list H-homomorphismproblem to bipartite graphs since there is no homomorphism of a non-bipartite graph to H.Brightwell [7] found the first proof that the general LORA problem with ` = 2 is NP–hard.Since his proof does not use Theorem 1, our proof turns out to be shorter and it coversmuch wider family of graphs than that of Brightwell.

Theorem 2. The general LORA problem with ` = 2 and with each cz = 0, and each costcz(u) in {0, 1} is NP–hard provided the complement of F is not a circular arc graph.

Proof: Let F be a bipartite graph and assume that the complement of F is not a circulararc graph (see Theorem 1). Let a bipartite graph G and lists Λ be an input of the listF -homomorphism problem. Define costs cz(u) for each z ∈ V (F ) and u ∈ V (G) as follows:cz(u) = 0 if z ∈ Λ(u) and cz(u) = 1, otherwise. We put cz = 0 for each z ∈ V (F ). In otherwords, the use of each vertex z ∈ V (F ) in homomorphisms of G to H is free. In this case,in the general LORA problem with ` = 2, we can always put L1 ∪ L2 = V (F ).

Let G1, G2, . . . , Gg be components of G and let F1, F2, . . . , Ff be components of F .Let Zj

1 , Zj2 be partite sets of Fj for every j = 1, 2, . . . , f. Observe that there exists a

Λ-homomorphism of G to F if and only if for each i = 1, 2, . . . , g there is a j(i) ∈{1, 2, . . . , f} such that there exists a Λ-homomorphism of Gi to Fj(i). However, there is

a Λ-homomorphism of Gi to Fj(i) if and only if the minimum cost of either a (Zj(i)1 , Z

j(i)2 )-

homomorphism of Gi to Fj(i) or a (Zj(i)2 , Z

j(i)1 )-homomorphism of Gi to Fj(i) is equal to

0 (with the costs defined above). Thus, we have a polynomial time Turing-reduction [13]from the NP–complete list H-homomorphism problem to the general LORA problem with` = 2. Hence, by the definition of the NP–hardness (see Section 5.1 in [13]), the generalLORA problem with ` = 2 is NP–hard. ut

It is well-known [15] (see also [16]) that for a fixed graph H, the problem to verifywhether there exists a homomorphism of an input graph G into H is NP–complete if H isnon-bipartite and polynomial time solvable if H is bipartite. Thus, the obvious extensionof the general LORA problem to non-bipartite graphs F is NP–hard.

3 LORA-M with ` = 2

Let B = (W1,W2;E) be a bipartite graph. For a vertex z ∈ W1 ∪ W2, let N(z) be theset of vertices adjacent to z. Orderings x1, x2, . . . , x|W1| and y1, y2, . . . , y|W2| of vertices ofW1 and W2, respectively, are called monotone if N(xi) ⊆ N(xi+1) and N(yj) ⊆ N(yj+1)for each i = 1, 2, . . . , |W1| − 1 and j = 1, 2, . . . , |W2| − 1. A bipartite graph B is calledmonotone if it has monotone orderings of its partite sets. Observe that if x1, x2, . . . , x|W1|

and y1, y2, . . . , y|W2| are monotone orderings, then xpyq ∈ E implies that xsyt ∈ E for eachs ≥ p and t ≥ q.

Notice that the bipartite graph FBR corresponding to the rules R1 and R2 of LORA-BRis monotone (consider orderings D,C, L and l, c, d), so are the bipartite graphs correspond-ing to R1 and R2 separately (there might be a situation when one of the rules is not used).Interestingly, monotone bipartite graphs form a family of so-called convex bipartite graphs;several families of convex bipartite graphs have been found useful in various applications,see [3].

Let B = (W1,W2; E) be a bipartite graph, let n = |W1| + |W2| and let m = |E|.One can test whether B is monotone in time O(m + n) as follows. Order vertices of W1

and W2 separately according to their degrees deg(z), x1, x2, . . . , x|W1| and y1, y2, . . . , y|W2|,such that deg(xi) ≤ deg(xi+1) and deg(yj) ≤ deg(yj+1) for each i = 1, 2, . . . , |W1| − 1and j = 1, 2, . . . , |W2| − 1. Observe that B is monotone if and only if these orderings aremonotone. We can use counting sort (see Chapter 9 of [8]) to get the orderings accordingto degrees in time O(n). The remaining computations can be carried out in time O(m).

The general LORA problem restricted to fixed monotone bipartite graphs F = (Z1, Z2; T )is called LORA-M. We assume that we have monotone orderings x1, x2, . . . , x|Z1| andy1, y2, . . . , y|Z2| of Z1 and Z2, respectively.

We reduce LORA-M to the maximal weight independent set problem on bipartitegraphs. Recall that a vertex set I of a graph is independent if there is no edge betweenvertices of I.

In the next theorem, we will consider a bipartite graph B with partite sets W1,W2 andnonnegative vertex weights p(u), u ∈ V (B), and the following (s, t)-network N (B): addnew vertices s and t to B, append all arcs su of capacity p(u), vt of capacity p(v) for allu ∈ W1 and v ∈ W2, and orient every edge xy of B, where x ∈ W1, from x to y (these arcsare of capacity ∞). For results on flows and cuts in networks see [8].

Theorem 3. If (S, T ) is a minimum cut in N (B), s ∈ S, then (S ∩W1) ∪ (T ∩W2) is amaximum weight independent set in B. One can find a maximum weight independent setin B in time O(n2

1

√m + n1m), where n1 = |U1| and m = |E(B)|.

The structural part of Theorem 3 is well-known, cf. Frahling and Faigle [12] (a similarresult is described in [17]). The complexity claim follows from the fact that one can find aminimum cut in N (B) in time O(n2

1

√m + n1m) by first finding a maximum flow by the

bipartite preflow-push algorithm of Ahuja et al. [2] and then finding a minimum cut (e.g.,by finding vertices reachable from s in the residual network using depth-first search).

Let us return to LORA-M and formulate it as a maximization problem. Choose setsLi ⊆ Zi, i = 1, 2. Let u ∈ Vi and set lists Λ(u) = Li, i = 1, 2. Recall that x1, x2, . . . , x|Z1| andy1, y2, . . . , y|Z2| are monotone orderings of Z1 and Z2. Assume that u ∈ V1, xp, xq ∈ Λ(u),p < q and cxp(u) > cxq(u). Observe that since cxp(u) > cxq(u) and F is monotone, anoptimal (L1, L2)-homomorphism k will not map u to xp. Thus, we may reduce the list Λ(u)of possible images of u by deleting xp. Certainly, we may reduce all Λ(v), v ∈ V1, such that

if xr, xs ∈ Λ(v) and r < s, then cxr(v) ≤ cxs(v). We call such a list Λ(v) reduced. Similarly,one defines the reduced list of a vertex in V2.

For a vertex u ∈ V , we can get the reduced list Λ(u) in time O(1) by the following simpleprocedure (the running time is constant since F is fixed). To simplicity the description,assume that u ∈ V1. The input is Λ(u) := L1 = {xp(1), xp(2), . . . , xp(t)}, p(1) < p(2) < · · · <p(t). We start from xp(t). We compare cxp(t)

(u) with cxp(t−1)(u), cxp(t−2)

(u),. . . and find themaximal i such that cxp(i)

(u) ≤ cxp(t)(u). We delete from Λ(u) all xp(i+1), xp(i+2), . . . , xp(t−1).

We compare cxp(i)(u) with cxp(i−1)

(u), cxp(i−2)(u),. . . and continue as above. Thus, we can

obtain the reduced lists Λ(v), v ∈ V , in time O(|V |).In the reminder of this section, we will use the following notation for the reduced lists:

Λ(u) = {zp(1), zp(2), . . . , zp(|Λ(u)|)}, where p(1) < p(2) < · · · < p(|Λ(u)|) and z = x if u ∈ V1

and z = y, otherwise.Recall that a homomorphism k of G to F is a Λ-homomorphism if k(u) ∈ Λ(u) for

each u ∈ V. Observe that LORA-M is equivalent to the problem of choosing sets Li ⊆ Zi,i = 1, 2 and finding a Λ-homomorphism k of G to F that minimize the cost of k, whereΛ(u) is the reduced list for u ∈ V.

Now we replace the costs by weights. Let M be the maximum of all costs in LORA-M(i.e., cz(u)’s and cz’s). For each pair of vertices z ∈ Zi and u ∈ Vi, i = 1, 2, let wz(u) =M − cz(u) and for each vertex z ∈ Z let wz = M − cz. Notice that, by the definition, allthe weights are nonnegative. Let k be a Λ-homomorphism of G to F. The weight of k isdefined as ∑

u∈V

wk(u)(u) +∑

z∈L1∪L2

wz. (2)

Observe that LORA-M is equivalent to the problem of choosing sets Li ⊆ Zi, i = 1, 2 andfinding a Λ-homomorphism k of G to F that maximize the weight of k, where Λ(u) is thereduced list for u ∈ V.

We now prove the following main result of the paper.

Theorem 4. For fixed subsets Li, i = 1, 2, LORA-M with ` = 2 can be solved in timeO(n2

1

√m + n1m + n), where n1 = |V1|, n = |V | and m = |E|.

Proof: Recall that all our graphs have no loops. If F is edgeless, then there is no homo-morphism of G to F. Thus, we may assume that x|U1|y|U2| ∈ T. Since Li, i = 1, 2, are fixed,for simplicity, we will assume that all weights wij = 0 in (2). Let Λ(u) be the reduced listfor each u ∈ V (we have shown how to find these lists in time O(n)).

Let W be a constant larger than max{wj(u) : u ∈ V, j ∈ Λ(u)}. Construct a new graphH with

∑u∈V |Λ(u)| vertices:

V (H) = {uz : u ∈ V, z ∈ Λ(u)}.

Let an edge uxvy be in H if uv ∈ E and xy 6∈ T . Let u ∈ V . For every j ∈ {1, 2, . . . , |Λ(u)|},let the weight w(uzp(j)

) be equal to wzp(j)(u) + W , if j = |Λ(u)|, and equal to wzp(j)

(u) −

wzp(j+1)(u), otherwise. Since each list Λ(u) is reduced, the weights of the vertices of H are

nonnegative.Clearly, if we replace, in G, a vertex u ∈ V by |Λ(u)| independent copies such that

there is an edge between a copy of u and a copy of v if and only if uv ∈ E, then we obtaina supergraph G∗ of H. Since G is bipartite, so is G∗ and, thus, H.

Observe that, by monotonicity of F , if uxp(i), uxp(j)

, vyp(f), vyp(g)

are vertices of H, j ≥ i,g ≥ f and uxp(i)

vyp(f)6∈ E(H), then uxp(j)

vyp(g)6∈ E(H) as well. We call this property of H

index-antimonotonicity.Assume that there exists a Λ-homomorphism k of G to F . Let k(u) = zp(iu). Then the

set {uzp(iu): u ∈ V } is independent in H. Moreover, by index-antimonotonicity of H,

S = ∪u∈V {uzp(j): iu ≤ j ≤ |Λ(u)|} (3)

is an independent set in H. Observe that S contains S′ = {uzp(Λ(u)): u ∈ V } and the

weight of S is equal to that of the homomorphism plus W × |V | (we use telescopic sums).Assume that a maximum weight independent set S in H contains S′. Then map each

u ∈ V to k(u) = zp(iu) such that iu = min{j : uzp(j)∈ S}. By maximality, S is of the

form (3) or, due to index-antimonotonicity of H, S may be extended to (3) by adding somevertices of zero weight. Observe that the weight of S is equal to that of the homomorphismplus W × |V |. If a maximum weight independent set S in H does not contain S′, then S′

is not an independent set in H (since the weight of S′ is larger than the weight of S) and,thus, there is no Λ-homomorphism of G to F .

Thus, there is an Λ-homomorphism of G to F if and only if a maximum weight in-dependent set in H contains S′. If there is an Λ-homomorphism of G to F , then thishomomorphism corresponds to a maximum weight independent set S in H. It remains toobserve that we may apply Theorem 3 to find a maximum weight independent set of H. ut

There are less than a = 2|Z1|+|Z2| choices of nonempty L1 and L2. Since F is fixed, a isa constant. Thus, we obtain the following:

Theorem 5. LORA-M with ` = 2 can be solved in time O(n21

√m + n1m + n), where n1,

n and m are defined in Theorem 4.

4 General LORA Problem and LORA-M

Let ` ≥ 2 be a constant. An `-partition X1, X2, . . . , X` of a set X is a collection of subsetsof X such that Xi ∩ Xj = ∅ for each i 6= j and X1 ∪ X2 ∪ · · · ∪ X` = X. An `-partitionX1, X2, . . . , X` of the vertex set X of a graph H is called layered if, for each edge xyof H, there exists an index i such that one vertex of xy is in Xi and the other is inXi+1. Observe that a graph H with a layered `-partition is bipartite with partite sets∪{Xi : 1 ≤ i ≤ `, i ≡ 1 (mod 2)} and ∪{Xi : 1 ≤ i ≤ `, i ≡ 0 (mod 2)}.

Let G = (V, E) be a graph with a layered `-partition V1, V2, . . . , V` of V . Let F = (U, T )be a fixed graph with a layered `-partition U1, U2, . . . , U` of U . Let Li ⊆ Ui, i = 1, 2, . . . , `.

We call a homomorphism k of G to F an (L1, L2, . . . , L`)-homomorphism of G to H ifk(u) ∈ Li for each u ∈ Vi, i = 1, 2, . . . , `.

We formulate the general LORA problem as follows: We are given a graph G as aboveand we consider homomorphisms k of G to F . Mapping u ∈ V to z ∈ U (i.e., k(u) = z)incurs a real cost cz(u). The use of a vertex z ∈ U in a homomorphism k (i.e., k−1(z) 6= ∅)incurs a real cost cz. We wish to choose subsets Li ⊆ Ui, i = 1, 2, . . . , `, and find an(L1, L2, . . . , L`)-homomorphism k of G to F that minimizes

∑

u∈V

ck(u)(u) +∑

z∈L

cz, (4)

where L = ∪`i=1Li. Notice that the graph F is fixed and is not part of the input.

By Theorem 2, the general LORA problem is NP–hard (even the general LORA problemin which all costs cz(u) = 0 for u ∈ Vi, i ≥ 3, is NP–hard). To define (the general) LORA-Mfor ` ≥ 2, let us define `-monotone graphs. Let F = (U, T ) be a fixed graph with a layered`-partition U1, U2, . . . , U`; F is called `-monotone if there is an ordering zi

1, zi2, . . . , z

i|Ui| of

vertices of Ui for each i = 1, 2, . . . , ` such that the subgraph F [Uj ∪ Uj+1] of F inducedby Uj ∪ Uj+1 is monotone with zj

1, zj2, . . . , z

j|Uj | and zj+1

1 , zj+12 , . . . , zj+1

|Uj+1| being monotoneorderings for each j = 1, 2, . . . , `−1. LORA-M is the general LORA problem with F being`-monotone. Similarly to Theorem 5, one can prove the following:

Theorem 6. LORA-M with fixed ` ≥ 2 can be solved in time O(n21

√m + n1m + n), where

n1 is the number of vertices in the smaller partite set of input graph G, n = |V (G)| andm = |E(G)|.

5 Minimum Cost H-Homomorphism Problem

This paper provides a motivation to study the following minimum cost H-homomorphismproblem (MCHP): For a fixed graph H and an input graph G given together with costscz(u), the cost of mapping a vertex u ∈ V (G) to z ∈ V (H), verify whether there is ahomomorphism of G to H, and if one exists, find such a homomorphism k that minimizes∑

u∈V (G) ck(u)(u).An argument similar to that in the proof of Theorem 2 shows that MCHP problem

generalizes the list H-homomorphism problem and that if H is not bipartite with thecomplement being circular arc graph, then MCHP is NP–hard.

Theorem 7. If H = (U1, U2, ; T ) is a monotone bipartite graph, then MCHP can be solvedin time O(n2√m + nm + n), where n is the number of vertices in the input graph G andm is the number of edges in G.

Proof: Let t(n,m) = O(n2√m + nm + n). Since H is bipartite (and loopless), if there isa homomorphism of G to H, then G is bipartite. So we may assume that G = (V1, V2; E)is bipartite.

Assume that G and H are connected. Then for each homomorphism k of G to H,we have either k(Vi) ⊆ Ui or k(Vi) ⊆ U3−i for every i = 1, 2. Thus, to find an optimalhomomorphism of G to H, it suffices to compute an optimal (U1, U2)-homomorphism andoptimal (U2, U1)-homomorphism and compare their costs. By Theorem 4, the total runningtime for finding the two optimal homomorphisms is t(n,m).

If H is disconnected, then by the definition of monotonicity, H consists of isolatedvertices and at most one component H ′, which is not an isolated vertex. The case when allcomponents of H are isolated vertices is trivial, so we may assume that H ′ does exist.

Assume that G consists of components G1, G2, . . . , Gb. Observe that every homomor-phism k of G to H consists of b ’independent’ homomorphisms ki : Gi→H. In fact, if Gi hasmore than one vertex that ki maps Gi into H ′ and, by the above, we can find an optimalhomomorphism of Gi to H ′ in time t(ni,mi), where ni = |V (Gi)| and mi = |E(Gi)|. If Gi

is a vertex v, ki may map it to any vertex of H and, in an optimal ki it maps Gi into z withminimum cz(u), z ∈ U1 ∪ U2. The running time to find such a vertex z is t(1, 0) = O(1).To complete our proof, it suffices to observe that

∑bi=1 t(ni,mi) = t(n, m). ut

The following theorem allows us to relate the NP-hardness and polynomial solvablecases above. Recall that a graph P = (V (P ), E(P )) is an interval graph if there is a familyof intervals Iv, v ∈ V (P ), of the real line, such that xy ∈ E(P ) if and only if Ix and Iy

intersect. The clique covering number of a graph B is the minimum number of completesubgraphs of B covering V (B).

Theorem 8. A graph H is a monotone bipartite graph if and only if its complement H̄ isan interval graph with clique covering number two.

Proof: First assume that H is a monotone bipartite graph with partite sets {v1, v2, . . . , vk}and {w1, w2, . . . , wl}. By the definition of a bipartite monotone graph we may assume thatviwj ∈ E(H) implies that vi′wj′ ∈ E(H) for all i′ ≥ i and j′ ≥ j. Let m(j) be defined asthe least index such that vm(j)wj ∈ E(H). Now consider the following intervals:

si = [i, k + 1] for all i = 1, 2, . . . , ktj = [0,m(j)− 1

2 ] for all j = 1, 2, . . . , l

Let B be the interval graph obtained from the above intervals, such that V (B) = S∪T ,where S = {s1, s2, . . . , sk} and T = {t1, t2, . . . , tl} and there is an edge between two verticesif and only if the corresponding intervals intersect. Note that both S and T form a cliquein B. Furthermore sitj ∈ E(B) if and only if i < m(j), which happens if and only ifviwj 6∈ E(H). Therefore B = H̄, and we have completed one direction.

So assume that H̄ is an interval graph with clique covering number two. Let [si, ti],i = 1, 2, . . . , k, denote the intervals corresponding to one of the cliques in the clique coverof size two and let [s′i, t

′i], i = 1, 2, . . . , l, denote the intervals corresponding to the other

clique in the clique cover. Let T denote the minimum value of all ti and let T ′ denote the

minimum value of all t′i. Without loss of generality we may assume that T ≤ T ′. Againwithout loss of generality we may assume that t1 ≥ t2 ≥ . . . ≥ tk and s′1 ≤ s′2 ≤ . . . ≤ s′l.

Assume that [si, ti] and [s′j , t′j ] do not intersect. Suppose that t′j < si, which implies

that tk < si contradicting the fact that [sk, tk] and [si, ti] intersect. Therefore we must haveti < s′j , which implies that [sa, ta] and [s′b, t

′b] do not intersect for any a ≥ i and b ≥ j.

Therefore H̄ is the complement of a monotone bipartite graph. utThe last two theorems imply the following:

Theorem 9. If H is a bipartite graph and its complement is an interval graph, then MCHPcan be solved in time O(n2√m + nm + n), where n is the number of vertices in an inputgraph G and m is the number of edges in G.

Let P5 be the path with 5 vertices. The graph P5 is not a monotone bipartite graph,but its complement is a circular arc graph. Thus, there remains a gap between the setof graphs H for which we showed that the problem is NP-hard and for which we provedthat it is tractable. It would be interesting to close the gap. We considered some directedextension of the 2-SAT approach of [11], but they did not appear to be useful.

6 LORA-BR and Critical Independent Set Problem

Let Q be an arbitrary graph. For a set X ⊆ V (Q), let N(X) = ∪x∈X{y ∈ V (Q) : xy ∈E(Q)}. Let p, q be a pair of functions from V (Q) to the set of nonnegative reals. In thecritical independent set problem (CISP) we seek A maximizing

{∑

a∈A

p(a)−∑

c∈N(A)

q(c) : A is an independent vertex set in Q}.

Clearly, CISP is NP–hard as the maximum weight independent set problem on arbitrarygraphs is CISP with q(u) = 0 for each u ∈ V (Q). Ageev [1] proved that CISP is polynomialtime solvable if p(u) = q(u) for each u ∈ V (Q). This generalized the corresponding resultof Zhang [19] for p(u) = q(u) = 1 for each u ∈ V (Q). We will show that CISP can be solvedin polynomial time on bipartite graphs for arbitrary functions p and q.

Theorem 10. CISP on a bipartite graph G = (V1, V2;E), V = V1 ∪ V2, can be solved intime O(n2

1

√m + n1m + n), where n1 = |V1|, n = |V | and m = |E|.

Proof: Observe that LORA-BR with fixed lists L1 = V1, L2 = V2 may be reformulatedas follows: Given a bipartite graph G = (V1, V2, E) and three weights wi(v), i = 1, 2, 3,for each vertex v ∈ V , we color every vertex of G in one of the colors 1,2,3 such that ifa vertex is colored 1, then all its neighbors must be colored 3. Assigning a color i to avertex v contributes weight wi(v) to the total weight of the coloring. We seek a coloring ofmaximum total weight.

Observe that if w1(u) < w2(u) for some u ∈ V, then there is an optimal coloring forwhich u is not colored 1. Thus, we may set w1(u) := w2(u) and keep a record, say (u, 1, 2),that indicates that if, in an optimal coloring that we found u is colored 1, we recolor it 2.Similar arguments allow us to assume that w1(u) ≥ w2(u) ≥ w3(u) for each u ∈ V.

Consider an optimal coloring, in which A is the set of vertices assigned color 1. Then A isindependent, all vertices of N(A) must have color 3 and all vertices of B = V (G)−A−N(A)may have color 2. The total weight of the coloring is

∑

a∈A

w1(a) +∑

c∈N(A)

w3(c) +∑

b∈B

w2(b) =∑

d∈V

w2(d)−∑

c∈N(A)

w2,3(c) +∑

a∈A

w1,2(a),

where w2,3(c) = w2(c)− w3(c), w1,2(a) = w1(a)− w2(a).Choose weight functions w1, w2, w3 as follows: w1(u) = p(u) + q(u), w2(u) = q(u),

w3(u) = 0 for each u ∈ V (G). Since∑

d∈V w2(d) is a constant, we observe that CISP onG (and functions p and q) can be reduced to LORA-BR with fixed L1 = V1, L2 = V2. Itremains to apply Theorem 4. ut

Acknowledgements We’d like to thank Graham Brightwell, David Cohen and MartinGreen for valuable discussions on the topic of the paper. Research of the first three authorswas partially supported by the Leverhulme Trust. Research of Gutin and Rafiey was sup-ported in part by the IST Programme of the European Community, under the PASCALNetwork of Excellence, IST-2002-506778.

References

1. A.A. Ageev, On finding critical independent and vertex sets, SIAM J. Discrete Math. 7 (1994), 293–295.2. R.K. Ahuja, J.B. Orlin, C. Stein, and R.E. Tarjan, Improved algorithms for bipartite network flows.

SIAM J. Comput. 23 (1994), 906–933.3. A.S. Asratian, T.M.J. Denley and R. Haggkvist. Bipartite Graphs and Their Applications. Cambridge

University Press, Cambridge, 1998.4. L. Barros, The optimisation of repair decisions using life-cycle cost parameters. IMA J. Management

Math. 9 (1988), 403–413.5. L. Barros, Private communications with M. Tso, 2001.6. L. Barros and M. Riley, A combinatorial approach to level of repair analysis, Europ. J. Oper. Res. 129

(2001), 242–251.7. G. Brightwell, Private communications with G. Gutin, Jan., 2005.8. T.H. Cormen, C.E. Leiserson and R.L. Rivest. Introduction to Algorithms. MIT Press, 1990.9. G. Cornuejols, G. L. Nemhauser, L. A. Wolsey. The Uncapacitated Facility Location Problem. Ch.3,

Discrete Location Theory, R. L. Francis and P. B. Mirchandani (Eds.). Wiley, New York, 1990.10. J.W. Crabtree and B.C. Sandel, 1989 Army level of repair analysis (LORA). Logistics Spectrum 1989

(Summer), 27–31.11. T. Feder, P. Hell and J. Huang. List homomorphisms and circular arc graphs, Combinatorica 19 (1999),

487–505.12. G. Frahling and U. Faigle, Combinatorial algorithm for weighted stable sets in bipartite graphs, to

appear in Discrete Appl. Math.13. M.R. Garey and D.S. Johnson. Computers and Intractability. Freeman and Co., San Francisco, 1979.14. B. Goldengorin. Requirements of standards: optimization models and algorithms. ROR Co., Hoogezand,

The Netherlands, 1995.15. P. Hell and J. Nesetril. On the complexity of H-coloring. J. Combin. Th. B 48 (1990) 92–110.16. P. Hell and J. Nesetril. Graphs and Homomorphisms. Oxford University Press, Oxford, 2004.17. D. Hochbaum, Provisioning, Shared Fixed Costs, Maximum Closure, and Implications on Algorithmic

Methods Today, Management Sci. 50 (2004), 709–723.18. D. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, N.J., 1996.19. C.-Q. Zhang, Finding critical independent sets and critical vertex subsets are polynomial problems,

SIAM J. Discrete Math. 3 (1990), 431–438.